Active sequential xampling receiver for spectrum sensing

ABSTRACT

An active sequential xampling receiver for spectrum sensing is disclosed. The receiver includes a dynamically adjustable analog front end to perform sub-Nyquist energy sensing across a broad radio frequency (RF) spectrum. In an exemplary aspect, the receiver includes a dynamic modulator which modulates the broad RF spectrum to dynamically select sub-bands (e.g., RF channels) and fold their spectral content into a narrower baseband signal for energy detection. A controller adjusts the dynamic modulator to maximize utility of the spectrum detection based on past energy observations.

RELATED APPLICATIONS

This application claims the benefit of provisional patent applicationSer. No. 62/619,291, filed Jan. 19, 2018, the disclosure of which ishereby incorporated herein by reference in its entirety.

FIELD OF THE DISCLOSURE

This application is related to radio frequency (RF) receivers, andspecifically to spectrum sensing in RF receivers.

BACKGROUND

Mobile devices are increasingly being used for entertainment, such asgaming and video streaming. The coexistence of these new services withthe Internet of Things (IoT) and machine-to-machine communications meansthat wireless applications may quickly become starved for bandwidth.Millimeter wave and other radio frequency (RF) spectrum can provide muchneeded increases in throughput but pose a challenge of high-speedsampling required to sense the spectrum. This may be a particularchallenge for relatively low-rate IoT applications, which are likely tobenefit from opportunistic decentralized spectrum access. To achieveefficient usage of the spectrum, sensing techniques are needed whichovercome the bottleneck of sampling at the Nyquist rate, which isgenerally too time and/or energy intensive, particularly for low-powerwireless devices.

Two classes of solutions have been proposed to overcome this bottleneck.The first approach uses a sub-Nyquist sampling front end usinganalog-to-digital conversion techniques named xampling architectures.Xampling architectures preprocess a signal in the analog domain and thensample at a lower rate compared with what the Nyquist theorem dictates.The aim is to reduce the complexity and energy cost for theanalog-to-digital converter hardware. However, prior xampling approachesincrease complexity in the reconstruction of the underlying signal, andaccurate spectrum sensing may not be guaranteed in lower signal-to-noiseratio (SNR) signals.

The second approach consists in selecting opportunistically, and in acognitive fashion, a small section of spectrum at a time, relying on ananalog front end able to switch between small sub-bands. This approachmay be better able to cope with lower SNR signals. However, for sensinga broad spectrum, the second approach may take too much time todetermine and use available spectrum before the environment has changed.

SUMMARY

An active sequential xampling receiver for spectrum sensing isdisclosed. The receiver includes a dynamically adjustable analog frontend to perform sub-Nyquist energy sensing across a broad radio frequency(RF) spectrum. In an exemplary aspect, the receiver includes a dynamicmodulator which modulates the broad RF spectrum to dynamically selectsub-bands (e.g., RF channels) and fold their spectral content into anarrower baseband signal for energy detection. A controller adjusts thedynamic modulator to maximize utility of the spectrum detection based onpast energy observations.

Nyquist sampling at very high carriers can be prohibitively costly forlow-power wireless devices. In spectrum sensing, this limit calls forthe receiver to include an analog front end that can sweep differentbands quickly in order to use the available spectrum opportunistically.Thus, the sensing action of the analog front end maximizes a utilityfunction decreasing linearly with the number of measurements andincreasing as a function of the active spectrum components that arecorrectly detected. The optimization selects the best linearcombinations of sub-bands to mix in order to accrue maximum utility. Thestructure of the utility represents the trade-off between exploration,exploitation, and risk of making an error that is characteristic of thespectrum-sensing problem.

An exemplary embodiment relates to an RF receiver. The RF receiverincludes a dynamic modulator configured to modulate a received signalsuch that one or more portions of the received signal are dynamicallyselected and folded into a baseband of a modulated signal. The RFreceiver also includes an energy detector configured to sample an energyof the modulated signal at a sub-Nyquist rate. The RF receiver alsoincludes a controller configured to adjust the dynamic modulator basedon an output of the energy detector.

Another exemplary embodiment relates to a method for sensing occupied RFspectrum. The method includes receiving an RF signal and sequentiallymodulating the RF signal using a sensing matrix to produce a set ofmodulated signals. The method also includes detecting a signal occupancyfor each of the set of modulated signals. In addition, coefficients ofthe sensing matrix are dynamically adjusted based on detecting thesignal occupancy.

Another exemplary embodiment relates to an analog front end for an RFreceiver. The analog front end includes a dynamic modulator configuredto convert an incoming RF signal having a first bandwidth to a modulatedsignal having content of the first bandwidth dynamically folded in asecond bandwidth narrower than the first bandwidth. The analog front endalso includes an energy detector configured to sense signal energy ofthe modulated signal within the second RF bandwidth.

Those skilled in the art will appreciate the scope of the presentdisclosure and realize additional aspects thereof after reading thefollowing detailed description of the preferred embodiments inassociation with the accompanying drawing figures.

BRIEF DESCRIPTION OF THE DRAWING FIGURES

The accompanying drawing figures incorporated in and forming a part ofthis specification illustrate several aspects of the disclosure, andtogether with the description serve to explain the principles of thedisclosure.

FIG. 1A is a schematic diagram of a cognitive radio scenario.

FIG. 1B is a graphical representation of a radio-frequency (RF) spectrumof the cognitive radio scenario of FIG. 1A.

FIG. 2 is a diagram of an exemplary receiver configured to sensespectrum at a sub-Nyquist rate by dynamically modulating portions of abroad spectrum.

FIG. 3 is a diagram of the exemplary receiver of FIG. 2, illustrating ingreater detail components of an analog front end.

FIG. 4A is a graphical representation comparing utility for differentoptimization approaches for a spectrum sensing application.

FIG. 4B is a graphical representation comparing utility for differentoptimization approaches for a RADAR application.

FIG. 5A is a graphical representation comparing utility for differentoptimization approaches versus a ratio horizon K over a number ofresources N with the horizon K equal to 10.

FIG. 5B is a graphical representation comparing utility for differentoptimization approaches versus the ratio horizon K over the number ofresources N with the horizon K equal to 30.

FIG. 6A is a graphical representation comparing utility for differentoptimization approaches versus a minimum signal-to-noise ratio (SNR)with the horizon K equal to 30.

FIG. 6B is a graphical representation comparing utility for differentoptimization approaches versus the minimum SNR with the horizon K equalto 10.

DETAILED DESCRIPTION

The embodiments set forth below represent the necessary information toenable those skilled in the art to practice the embodiments andillustrate the best mode of practicing the embodiments. Upon reading thefollowing description in light of the accompanying drawing figures,those skilled in the art will understand the concepts of the disclosureand will recognize applications of these concepts not particularlyaddressed herein. It should be understood that these concepts andapplications fall within the scope of the disclosure and theaccompanying claims.

It will be understood that, although the terms first, second, etc. maybe used herein to describe various elements, these elements should notbe limited by these terms. These terms are only used to distinguish oneelement from another. For example, a first element could be termed asecond element, and, similarly, a second element could be termed a firstelement, without departing from the scope of the present disclosure. Asused herein, the term “and/or” includes any and all combinations of oneor more of the associated listed items.

It will be understood that when an element is referred to as being“connected” or “coupled” to another element, it can be directlyconnected or coupled to the other element or intervening elements may bepresent. In contrast, when an element is referred to as being “directlyconnected” or “directly coupled” to another element, there are nointervening elements present.

The terminology used herein is for the purpose of describing particularembodiments only and is not intended to be limiting of the disclosure.As used herein, the singular forms “a,” “an,” and “the” are intended toinclude the plural forms as well, unless the context clearly indicatesotherwise. It will be further understood that the terms “comprises,”“comprising,” “includes,” and/or “including” when used herein specifythe presence of stated features, integers, steps, operations, elements,and/or components, but do not preclude the presence or addition of oneor more other features, integers, steps, operations, elements,components, and/or groups thereof.

Unless otherwise defined, all terms (including technical and scientificterms) used herein have the same meaning as commonly understood by oneof ordinary skill in the art to which this disclosure belongs. It willbe further understood that terms used herein should be interpreted ashaving a meaning that is consistent with their meaning in the context ofthis specification and the relevant art and will not be interpreted inan idealized or overly formal sense unless expressly so defined herein.

An active sequential xampling receiver for spectrum sensing isdisclosed. The receiver includes a dynamically adjustable analog frontend to perform sub-Nyquist energy sensing across a broad radio frequency(RF) spectrum. In an exemplary aspect, the receiver includes a dynamicmodulator which modulates the broad RF spectrum to dynamically selectsub-bands (e.g., RF channels) and fold their spectral content into anarrower baseband signal for energy detection. A controller adjusts thedynamic modulator to maximize utility of the spectrum detection based onpast energy observations.

Nyquist sampling at very high carriers can be prohibitively costly forlow-power wireless devices. In spectrum sensing, this limit calls forthe receiver to include an analog front end that can sweep differentbands quickly in order to use the available spectrum opportunistically.Thus, the sensing action of the analog front end maximizes a utilityfunction decreasing linearly with the number of measurements andincreasing as a function of the active spectrum components that arecorrectly detected. The optimization selects the best linearcombinations of sub-bands to mix in order to accrue maximum utility. Thestructure of the utility represents the trade-off between exploration,exploitation, and risk of making an error that is characteristic of thespectrum-sensing problem.

Compared with other multi-band signals receivers, the hardware in thearchitecture according to the present disclosure is simpler, since thearchitecture uses a single non-coherent receiver and a single samplingdevice that collects energy measurements sequentially, sampling at afraction of the Nyquist rate. A time-dependent utility function is usedto optimize the trade-off between sensing and exploitation.

It is important to remark that the optimum action generally does notattempt the full recovery of all the white spaces. The optimum decisionmay be conservative and sense a very limited portion of the spectrum.Furthermore, since scanning one sub-band at a time is a possible actionof the active spectrum-sensing strategy, the method of the presentdisclosure subsumes previous techniques to scan the spectrum optimally,without mixing it. This approach is referred to as direct inspection andis discussed in Section III, A.

The present disclosure is more closely related to the stochasticoptimization schemes that extend the framework and optimize acompressive spectrum-sensing action based on previous observations. Thecommon goal of stochastic optimization schemes is the recovery of thefull support of a given vector. Typically, such techniques are shown tobe able to cope with lower signal-to-noise ratio (SNR) in the signalreconstruction with low complexity. What these stochastic optimizationschemes do not capture is that, in cognitive spectrum-sensingapplications, a timely decision is also desirable to have enough time toexploit the spectrum. The method of the present disclosure is alsoadaptive with respect to the time horizon K, the number of resources N,the prior probabilities on the states of each resources, and theparameters that characterize the utility function, that is, reward andpenalty for good/bad decisions.

Based on performance analysis, it is the clear that sparse and adaptivesensing matrix designs outperform dense sensing matrices and those thatare sparse but static. There are two main reasons for this: (1) throughbelief propagation algorithms, they achieve near-optimum detectionperformance, and (2) they mitigate the aforementioned noise-foldingphenomena. The method of the present disclosure is applicable not onlywhen the utility comes from finding empty entries (e.g., spectrumsensing) but also when one is interested in finding the occupied ones(e.g., in a RADAR application).

More specifically, the present disclosure is organized as follows:Section I is dedicated to the signal model and the analog front end ofthe detector according to the present disclosure, and Section IIformulates the optimization problem. Then, Section III addresses theoptimal dynamic design for the direct inspection case (III, A), in whichthere is no mixing of sub-bands (also known as scanning receiver), and agroup testing case (III, B) in which is introduced the possibility ofmixing different bands. Section IV is dedicated to additionaldiscussions: an alternative standard method for detection based onmaximum likelihood estimate and LASSO relaxation, hardware limitationsof the architecture according to the present disclosure, and theextension of the framework according to the present disclosure to otherapplications. The present disclosure demonstrates that, even if findingthe optimal policy is exponentially complex in the number of resources,the approximation factor for a greedy procedure can be characterized.Numerical results to sustain the claims of the present disclosure arepresented in Section V.

Regarding notation, bold lowercase represents vectors, bold uppercaserepresent matrices, and calligraphic letters indicate sets. For example,s_(A) indicates the entries i∈

of vector s, and ∥y∥_(A) ² represents the weighted

₂-norm y^(T) Ay. For any set function ƒ (

), the marginal increment for adding element a is defined as ∂_(a)ƒ(

)=ƒ(

+a)−ƒ(

).

I. Signal Model and Receiver Overview A. Cognitive Radio Scenario

FIG. 1A is a schematic diagram of a cognitive radio scenario 10. Thecognitive radio scenario 10 includes a number of RF devices, which mayinclude one or more user equipment (UE) 12, base stations 14, andsimilar devices. Each of the UEs 12 and base stations 14 may send and/orreceive signals over a common RF spectrum. Accordingly, a receiver 16 inthe cognitive radio scenario 10 may need to quickly and accurately sensewhich portions of the RF spectrum are occupied in order to receiveand/or transmit signals within the RF spectrum as needed.

FIG. 1B is a graphical representation of an RF spectrum 18 of thecognitive radio scenario 10 of FIG. 1A. The RF spectrum 18 may bespectrum with opportunistic access, such as millimeter wave (60-80 GHz),citizens broadband radio service (CBRS), or other frequencies. The RFspectrum 18 being sensed by the receiver 16 can be represented in thefrequency domain as X(ƒ) and in the time domain as x(t). In addition,there may be N channels 20 (e.g., sub-bands centered at frequencies ƒ₁,ƒ₂, . . . , ƒ_(i), . . . , ƒ_(N)) of the RF spectrum 18, each channel 20having a channel width R_(c) such that the RF spectrum 18 has abandwidth of W=NR_(c).

In the context of cognitive radio for the RF spectrum 18, eachtransmission includes large amounts of control signals overhead inaddition to the payload. Therefore, it can be assumed that the activityof the UEs 12 and base stations 14 over one or more channels 20 willpersist for several sampling periods T_(s). However, assuming thisinterval T lasts for a multiple K of the sampling period T_(s) (i.e.,T=KT_(s)), the sensing mechanism of the receiver 16 should provide thefastest decision it can. The goal of the receiver 16 according to thepresent disclosure is to sequentially sense the spectrum for a firstportion of this interval T and transmit the most it can during theremaining time over the channels 20 found empty (or, alternatively,process one or more occupied channels 20).

B. Spectrum Sensing Architecture

FIG. 2 is a diagram of an exemplary receiver 16 configured to sensespectrum at a sub-Nyquist rate by dynamically modulating portions of abroad spectrum. The receiver 16 includes an analog front end 22, whichincludes an energy detector 24 and a dynamic modulator 26. The energydetector 24 is configured to sample energy of an input signal at asub-Nyquist rate. In an exemplary aspect, the energy detector 24 is asingle-channel detector which can operate quickly with low power demandsby filtering and sampling a single channel baseband input signal.

The dynamic modulator 26 facilitates use of the single channel energydetector 24 by modulating a received signal x(t) (e.g., the RF spectrum18) such that portions (e.g., sub-bands or channels 20) of the receivedsignal x(t) are dynamically selected and folded into the single channelbaseband input signal of the energy detector 24. The portions of thereceived signal x(t) which are selected and folded into the basebandinput signal by the dynamic modulator 26 are selected by a controller 28in order to maximize utility of the spectrum sensing function asdescribed further below in Sections II and III. The controller 28 cancontrol the dynamic modulator 26 through a sensing matrix B or othercontrol signals.

The receiver 16 can thus perform carrier sensing over multiple bandssimultaneously. The receiver 16 has the advantage of being sequential,and not requiring time accurate time offset between sampling channel asin some approaches. In addition, the spectrum sensing function of thereceiver 16 is robust to the noise-folding problem, in which SNRdeteriorates approximately linear in the number of bands that arealiased prior to sampling.

FIG. 3 is a diagram of the exemplary receiver 16 of FIG. 2, illustratingin greater detail components of the analog front end 22. In the proposedanalog front end 22, the dynamic modulator 26 folds the spectrum presentin specific sub-bands onto the center frequency of the energy detector24 for filtering and sampling, during what can be referred to as asub-Nyquist carrier sensing phase. The samples are spaced by intervalsof duration

$T_{s} = \frac{1}{R_{s}}$

which is a factor 1/N smaller than the total spectrum. As the diagram inFIG. 3 shows, rather than having a filter bank architecture, the analogfront end 22 performs sequential non-coherent tests. These tests aredesigned according to a utility-maximizing strategy, such as with thesensing matrix B chosen by the controller 28 of FIG. 2 via a greedyalgorithm, and the tests' thresholds y (described further below inSections II and III). It is assumed that the complex envelope of theanalog signal we are exploring is a multicomponent signal, whosecomponents are a frequency band width equal to W=NR_(s). During theinterval 0≤t<T the received signal is:

$\begin{matrix}{{y(t)} = {{x(t)} + {w(t)}}} & {{EQ}.\mspace{14mu} 1} \\{{x(t)} = {\sum\limits_{i = 1}^{N}\; {s_{i}{x_{i}(t)}e^{{- j}\; 2\; \pi \; {R_{s}{({i - 1})}}^{t}}}}} & {{EQ}.\mspace{14mu} 2}\end{matrix}$

with

(t)˜

(0, N₀δ(τ)) being additive white Gaussian noise. The components of thereceived signals x_(i)(t) correspond to each primary user source,modeled as band-limited random processes with bandwidth R_(s); they areequal in the mean square sense to the following process:

$\begin{matrix}{{x_{i}(t)} = {\sum\limits_{k = 1}^{N}\; {{x_{i}\lbrack k\rbrack}{{{sinc}\left( {\pi \left( {{R_{s}t} - k + 1} \right)} \right)}.}}}} & {{EQ}.\mspace{14mu} 3}\end{matrix}$

The dynamic modulator 26 modulates the received signal y(t) over theperiod (k−1)T_(s)≤t<kT_(s) with:

β_(k)(t)=Σ_(i=1) ^(N)√{square root over (b _(ki))}e ^(j(2πR) ^(s)^((i−1)t+ϕ) ^(i) ⁾.  EQ. 4

to produce a modulated signal, where b_(ki) are the coefficients of thesensing matrix B and the phase ϕ_(i) accounts for the delay ingenerating the tone at the ith frequency plus the oscillator phase. Inthis regard, the dynamic modulator 26 can be implemented with a set of Lvoltage controlled oscillators (VCOs) 30. The sensing matrix B cantherefore control a voltage level generation network 32 for the inputsto the VCOs 30 (and/or gains 34). The VCOs 30 are not required to besynchronized or phase-locked. Furthermore, the switching frequency ofthe VCOs 30 may also be R_(s) (i.e., the single channel bandwidth).

The energy detector 24 receives and samples an energy of the modulatedsignal. In an exemplary aspect, the energy detector 24 includes alow-pass filter (LPF) 36, which is convolved with the modulated signal.The LPF 36 may have an impulse response sinc(πR_(s)t) and outputs thefiltered modulated signal c(t). The energy detector 24 also includes asampling circuit 38, which then samples the filtered modulated signalc(t) at times kT_(s); k=1, . . . ,

in the single-channel baseband. This operation can be consideredequivalent to an orthogonal projection, as shown below:

$\begin{matrix}{{c\lbrack k\rbrack} = {\left. {\left\lbrack {{y(t)}{\beta_{k}(t)}} \right\rbrack*R_{s}{{sinc}\left( {\pi \; R_{s}t} \right)}} \right|_{t = {kT}_{s}} = {\sum\limits_{i = 1}^{N}\; {\sqrt{b_{ki}}e^{j\; \varphi_{i}}Y_{ki}}}}} & {{EQ}.\mspace{14mu} 5}\end{matrix}$

where Y_(ki) represents the orthogonal projections over the period(k−1)T_(s)≤t<kT_(s) of y(t) over the following signals:

Y _(ki) =

y(t),R _(s) e ^(j2πR) ^(s) ^((i−1)t) sinc(π(R _(s) t−k+1))

.  EQ. 6

If the periodic signals were not truncated in time, the relationshipwould be exact; in practice, however, there are some approximationerrors due to the windowing of the signal over the prescribed interval[(k−1)T_(s), kT_(s)]. The effect of this can be mitigated by usingraised cosine filtering and a non-rectangular window to reduce theeffect of side lobes. Considering that the signals

{e ^(j2πR) ^(s) ^((i−1)t) sinc(π(R _(s) t−k+1))}_(i,k∈Z)

form an orthogonal basis, and that Equation 1 is equivalent to thefollowing:

$\begin{matrix}{{x(t)} = {\sum\limits_{k = 1}^{K}\; {\sum\limits_{i = 1}^{N}\; {s_{i}{x_{i}\lbrack k\rbrack}e^{j\; 2\pi \; {R_{s}{({i - 1})}}t}{{sinc}\left( {\pi \left( {{R_{s}t} - k + 1} \right)} \right)}}}}} & {{EQ}.\mspace{14mu} 7} \\{Y_{ki} = {{s_{i}{x_{i}\lbrack k\rbrack}} + {w_{i}\lbrack k\rbrack}}} & {{EQ}.\mspace{14mu} 8}\end{matrix}$

where w_(i)[k]˜C

(0; n_(i)). If x_(i)[k] is also modelled as i.i.d. x_(i)[k]˜C

(0; φ_(i)), then for a given state s:

Y _(ki) ˜C

(0,φ_(i) +N ₀),  EQ. 9

where φ is a vector collecting the average, unknown a priori, receivedsignal power from the existing communications. The energy detector 24samples for k=1, . . . , κ are as follows:

$\begin{matrix}{{c\lbrack k\rbrack} = {\sum\limits_{i = 1}^{N}\; {\sqrt{b_{ki}}{e^{j\; \varphi_{i}}\left( {{s_{i}{x_{i}\lbrack k\rbrack}} + {w_{i}\lbrack k\rbrack}} \right)}}}} & {{EQ}.\mspace{14mu} 10}\end{matrix}$

and therefore (assuming ϕ_(i)'s are independent and uniformlydistributed in [0, 2π)) they are also conditionally zero mean Gaussianrandom variables:

c[k]˜C

(0,θ[k]),  EQ. 11

θ[k]=θ(b _(k) ,s)

b _(k) ^(T)(φ+n)  EQ. 12

It follows that, with reference to FIGS. 1A and 1B, the information forthe detection of communications by the UEs 12 and/or base stations 14 isembedded in the variance of the sample, which is the energy receivedduring the kth period. Sufficient statistics for the problem are asfollows:

y[k]

|c[k]|²  EQ 13

which are exponentially distributed, that is, y[k]˜Exp(θ[k]).

C. Hardware Considerations

The circuit diagram of FIG. 3 assumes a settling time for the VCOs 30much smaller than T_(s) (e.g., the sampling period for the singlechannel sub-band). If this assumption does not hold, one should use alow-pass filter with a smaller bandwidth and collect the samples c[k] atan even slower rate than R_(s) to wait for the VCOs to settle. Thismodification does not alter the statistical characterization of thesamples, derived in subsection I.B above. The drawback of taking samplesless often is that (assuming the same occupancy coherence time) one hasaccrued less information than what is available in the received signaland has less than K slots to decide. Given that the strategy is derivedas a function of K, this does not invalidate the findings. Anotherpossibility is to replace the L tunable VCOs 30 with N oscillators atconstant frequencies, corresponding to the N possible bands of thesignal. Using N oscillators increases the power consumption and cost ofthe circuit but significantly reduces the switching time between twomeasurements. Hence, this is the natural choice if one wants to exploita dense sensing matrix. Instead, the use of a set of VCOs 30 ispreferable if the matrices are sparse because a small number of VCOs cansynthesize the mixing signal. The switching is performed by amultiplexer that takes the sum of the up to L tones selected by thevector b_(k).

In general, since the detection of the signal is the focus, withreasonably good device components, calibration can be expected to beeither far less demanding or unnecessary, if one accepts loss insensitivity. The binary coefficients for the vector b can be set to onesand zeros, as discussed in Section III, B. Controlling the gains 34 isunnecessary for the system to work, and it may be preferable not to addtunable gains 34 because they can be another possible source ofuncertainty and complexity in the system. Finally, imperfect tuning ofthe VCOs 30 reduces the SNR, either by spreading or misplacing thecenter frequency of the components of interest, but not fundamentallyimpairing its detection.

II. Optimization Framework

With reference to FIGS. 1A-3, the receiver 16 can be modeled as needingto divide K instants of time available between the sensing and theexploitation of a set

={1; 2, : : : , N} of sub-bands (e.g., channels 20 of the RF spectrum18). A utility function of the receiver 16 accrues a reward that is afunction of the underlying state vector for these items:

s

[s ₁ , . . . ,s _(N)]∈{0,1}^(N)  EQ. 14

where the entries s_(i)∈{0,1}, as well as the residual time availableafter sensing. The state variables s_(i) are indicators of good/bad(0=1) state of a resource; for the cognitive spectrum sensingapplication 0 would mean the channel is “idle” and 1 would mean thechannel is “busy”. The receiver 16 acquires information about theentries via random observations coming from a known probability densityfunction parameterized by an unknown vector.

During the times devoted to sensing k=1, 2, . . . ,

<K the receiver 16 can dynamically and adaptively design eachmeasurement by selecting a subset of entries of s to probe through asensing vector b_(k)=[b_(k1), b_(k2), . . . , b_(kN)]. The b_(ki)'s willbe non-zero only on the channels that are actively sensed; it is assumedthe measurement provides an observation y[k] drawn from a densityƒ_(θ[k])(y) that is a function of both the choice of b_(k) and the states, i.e. θ[k]=θ(b_(k), s). It is assumed the state variables s_(i) aremutually independent Bernoulli random variables with known priorprobabilities given by a vector ω=[ω₁, ω₂, . . . , ω_(N)] whereω_(i)=P(s_(i)=0). As described further in Section III below, thereceiver can be designed with:

-   -   1) the        ×N measurement matrix B (exploration phase)    -   2) a set of N decision rules δ={δ_(i)∈{0,1}: i=1, 2, . . . , N}        over the unknown states s_(i) of the resources (at the end of        the exploration phase).        Notice that the design of B includes:    -   the measurement vectors b_(k) for each test at time k=1, 2, . .        . ,        (matrix rows),    -   the sensing (exploration) time        to acquire information on the states s_(i) via the observations        y[k] (number of rows).

The total utility for the receiver 16 will be proportional to the timeleft for exploitation (K−k). A reward r_(i)>0 can be used for correctlydetecting an empty/busy resource and a penalty p_(i)<0 for failing todetect a busy/empty resource. The utility of the detection function canbe written as:

$\begin{matrix}{{U\left( {s,,K,B,\delta}\; \right)}\overset{\Delta}{=}\left\{ \begin{matrix}{{\left( {K - k} \right){\sum\limits_{i = 1}^{N}\; {\omega_{i}{r_{i}\left( {1 - \alpha_{i}} \right)}}}} + {\left( {1 - \omega_{i}} \right)p_{i}\beta_{i}}} & {{case}\mspace{14mu} 0} \\{{\left( {K - k} \right){\sum\limits_{i = 1}^{N}\; {\left( {1 - \omega_{i}} \right)\left( {1 - \beta_{i}} \right)r_{i}}}} + {\omega_{i}\alpha_{i}p_{i}}} & {{case}\mspace{14mu} 1}\end{matrix} \right.} & {{EQ}.\mspace{14mu} 15}\end{matrix}$

where α_(i); β_(i) denote the type I and type II errors probabilityrespectively (e.g., α_(i)=P(δ_(i)=1|s_(i)=0) and β_(i)=P(δ_(i)=0|s_(i)=1).

To differentiate between case 0 and case 1 allows to considerapplications where the utility comes from an action on the entriesdetected as empty/busy: e.g., in a spectrum sensing application, theutility would come from the decision on transmitting over frequencybands found empty, whereas for a RADAR application, it makes more senseto consider the utility comes from taking action on the frequency(spatial directions) found busy. Finding the optimal policy correspondsto solving the following optimization problem:

max_(B,δ)

[U(s,

,K,B,δ)]  EQ. 16

This approach can be modified to fit different observation models andassumptions, but as an example this disclosure considers the followingform for the function θ:

θ[k]=θ(b _(k) ,s)=b _(k)(φ^(T) +w ^(T))  EQ. 17

where φ=[φ₁, φ₂, . . . , φ_(N)] is a non-negative vector, such that thestate variable s_(i)=1 when φ_(i)>0 and 0 when φ_(i)=0. The vectorw=[w₁, w₂, . . . , w_(N)] models a generic additive system noise. Anexemplary aspect of this disclosure concerns the detection of thenon-negative entries of φ and consequently the maximization of theutility accruable from the resources declared to be in the empty/busystate (Equation 15). The observation model also assumes thatƒ_(θ[k])(y)≡Exp(θ[k]), i.e.:

$\begin{matrix}{{{y\lbrack k\rbrack} \sim {f_{\theta {\lbrack k\rbrack}}(y)}} = {\frac{1}{\theta \lbrack k\rbrack}e^{- \frac{y}{\theta {\lbrack k\rbrack}}}}} & {{EQ}.\mspace{14mu} 18}\end{matrix}$

where, for convenience, the alternative parameterization for theexponential distribution is generally used herein. In the context of RFspectrum sensing, this choice of distribution models the energy ofcomplex circularly symmetric signal samples in each sub-band with zeromean and variance φ_(i) embedded in additive white Gaussian noise withvariance w₁.

III. Dynamic Design of Sensing Matrices A. Direct Inspection Case

In the direct inspection (DI) case, b_(k) is limited to only onenon-zero entry i, that is, b_(ki)≠0, b_(kj)=0 ∀j≠i. This means thatthere is an underlying hypothesis testing:

₀ : y[k]˜Exp(θ₀[k])

₁ : y[k]˜Exp(θ[k])

with θ₀[k]=b_(ki)n_(i) and θ[k]=b_(ki)(φ_(i)+n_(i))>θ₀[k]. In thiscontext, it is known that the signal energy is a sufficient statisticfor the test and the energy detection is optimal. Assuming no priorknowledge over the φ_(i)'s in case of existing communication, to set thetest threshold alone is needed, which is set in order to maximize theutility defined in Equation 15. Defining θ*[k]=max{y[k],b_(ki)(φ_(min)+ω_(i))} obtains the following:

$\begin{matrix}{{y\lbrack k\rbrack}_{\begin{matrix} < \\H_{0}\end{matrix}}^{\begin{matrix}H_{1} \\ > \end{matrix}}\frac{\ln \left( {\gamma_{i}\frac{\theta*\lbrack k\rbrack}{\theta_{0}}} \right)}{\frac{1}{\theta_{0}} - \frac{1}{\theta*\lbrack k\rbrack}}} & {{EQ}.\mspace{14mu} 19} \\{\gamma_{i}\overset{\Delta}{=}\left\{ \begin{matrix}\frac{r_{i}\omega_{i}}{{\rho_{i}}\left( {1 - \omega_{i}} \right)} & {{case}\mspace{14mu} 0} \\\frac{\left. {{\rho_{i}}\omega_{i}} \right)}{r_{i}\left( {1 - \omega_{i}} \right)} & {{case}\mspace{14mu} 1}\end{matrix} \right.} & {{EQ}.\mspace{14mu} 20}\end{matrix}$

Notice that, assuming a minimum average received signal power ofφ_(min)>0 in the case of existing transmission, this makes the testmeaningful also for values of γ_(i)<1.

Assumption 1:

To simplify the decision problem, every resource is assumed to be sensedbefore being declared empty/busy. This can be enforced as astandard/protocol rule or numerically guaranteed by setting ∀i∈

,

$\omega_{i} < {\frac{\rho_{i}}{\rho_{i} + r_{i}}\mspace{11mu} {\left( {{case}\mspace{14mu} 0} \right)/\omega_{i}}} > {\frac{r_{i}}{{\rho_{i}} + r_{i}}\mspace{11mu} {\left( {{case}\mspace{14mu} 1} \right).}}$

It is clear that the optimality of the test completely characterizes theset of decision rules δ for the sensed resources, while Assumption 1gives the decision rule for the non-sensed resources. This implies thatfor the DI case, the optimization in Equation 16 can be expressed solelyin terms of B. It is also known that for this type of text, where thereis uncertainty in a parameter of the alternative hypothesis, one doesnot know the exact miss probability β, thus an upper-bound is used,which reflects in a lower bound for the achievable utility. Since thistest is part of the DI strategy, the superscript ^(DI) is added to thetest error probabilities α_(i) and β_(i) as follows:

$\begin{matrix}{\alpha_{i}^{DI} = {\min \left\{ {\left( \frac{{\rho_{i}}\left( {1 - \omega_{i}} \right)}{r_{i}{\omega_{i}\left( {1 + \frac{\phi \; \min}{n_{i}}} \right)}} \right)^{\frac{1 + \frac{\phi \; \min}{n_{i}}}{\frac{\phi \; \min}{n_{i}}}},1} \right\}}} & {{EQ}.\mspace{14mu} 21} \\{\beta_{i}^{DI} = {1 - {\left( \alpha_{i}^{DI} \right)^{\frac{1}{1 + \frac{\phi_{i}}{n_{i}}}}.}}} & {{EQ}.\mspace{14mu} 22}\end{matrix}$

The above shows that the false alarm probability is independent from thealternative hypothesis, whereas the detection improves with the trueaverage transmitted power φ_(i). What can be guaranteed, sinceφ_(i)≥φ_(min), is that

$\begin{matrix}{{\beta_{i}^{DI} \leq {1 - \left( \frac{\rho_{i}\left( {1 - \omega_{i}} \right)}{r_{i}{\omega_{i}\left( {1 + \frac{\phi_{\min}}{n_{i}}} \right)}} \right)^{\frac{n_{i}}{\phi \; \min}}}} = \beta_{i,\max}^{DI}} & {{EQ}.\mspace{14mu} 23}\end{matrix}$

The threshold in Equation 20 is the optimal threshold that minimizes theBayesian risk (maximizes the utility) for the binary case, when ω_(i) isknown. It is common practice to replace the maximum likelihood estimatefor the unknown ω_(i) (Generalized Likelihood Ratio Test (GLRT)) andthen reduce to the binary case using the same threshold. A local, morepowerful test exists for θ→θ₀, but GLRT is preferred for its high SNRrange.

Note that the test performance for the DI case does not depend onb_(ki); therefore, for the DI case no further optimization is neededover the sensing matrix B other than selecting the non-zero entries.

Under Assumption 1, the optimization problem in Equation 16 can berewritten for the DI case as follows:

$\begin{matrix}{\underset{ \subseteq }{maximize}\mspace{11mu} {U^{DI}()}} & {{EQ}.\mspace{14mu} 24} \\{{U^{DI}()}\overset{\Delta}{=}{\left( {K - {}} \right){\sum\limits_{i \in A}\; u_{i}^{DI}}}} & {{EQ}.\mspace{14mu} 25} \\{u_{i}^{DI}\overset{\Delta}{=}{{\omega_{i}{r_{i}\left( {1 - \alpha_{i}^{DI}} \right)}} - {\left( {1 - \omega_{i}} \right)\rho_{i}\beta_{i,\max}^{DI}}}} & {{EQ}.\mspace{14mu} 26}\end{matrix}$

The following lemma is then introduced:

Lemma 1:

U^(DI)(

) is a normalized, non-monotone, non-negative sub-modular function of

.

Lemma 1 implies that there are diminishing returns in augmenting sets byadding a certain action to bigger and bigger sets. The maximization of anon-monotonic sub-modular function is generally NP-hard, but the case ofinterest is not as difficult. By sorting the resources i so that

u ₁ ^(DI) ≥u ₂ ^(DI) ≥ . . . ≥u _(N) ^(DI)  EQ. 27

the set of size i,

_(i)={1, . . . , i} will be such that for any set X of size |X|=i, sothat

${\sum\limits_{k = 1}^{i}\; u_{k}^{DI}} \geq {\sum\limits_{k \in X}^{\;}\; {u_{k}^{DI}.}}$

Therefore, what remains is to find the best set size i such that

$\begin{matrix}{{{U^{DI}()} \leq {U^{DI}\left( _{i} \right)} \leq_{i}^{\max}\left( {\left( {K - i} \right){\sum\limits_{k = 1}^{i}\; u_{k}^{DI}}} \right)}\mspace{14mu}} & {{EQ}.\mspace{14mu} 28}\end{matrix}$

The maximum in Equation 28 is attained for the following:

i*= _(i) ^(inf) {i: ∂ _(i+1) U ^(DI)(

_(i))<0}  EQ. 29

where

∂_(i+1) U ^(DI)(

_(i))=(K−i)u _(i+1) ^(DI)−Σ_(k=1) ^(i+1) u _(k) ^(DI).

Given that the function is sub-modular, as soon as this condition isattained, it is maintained for i+2, i+3, and so on, given that themarginal returns continue to decrease. This maximization is greedy andstops when the marginal reward becomes negative.

B. Group Testing Approach

The test is now allowed to mix different sub-bands, that is, the vectorb_(k) to have more than one non-zero entry. The main idea of thissection is to develop a dynamic simple strategy that can becharacterized in closed form, and gives a sufficient condition to claima group testing (GT) strategy would outperform the DI alternative. Fromthe sensing matrix B, the sets

_(k)={i∈

: b_(ki)≠0} and

_(i)={1≤k≤

: b_(ki)≠0} can be defined. The convention is used that, whenever b_(k)is the argument of a function, then the set

_(k) is also the argument of that function. Similarly, when B is theargument of a function, then all the sets

_(k) for 0≤k≤

−1 and all the sets

_(i) for i∈

are arguments of that function as well.

As outlined previously, aliasing of the spectrum comes with anassociated noise-folding phenomenon. The effect is particularly severein a non-coherent scheme such as that of the present disclosure. Thesamples are collected sequentially and not in parallel, which means thatthere are no multiple observations of the same value but only sequentialobservations tied to the same underlying random process.

To mitigate the noise-folding effects and reduce the hardwarecomplexity, the method of the present disclosure focuses on low-densitymeasurement matrices. The goal is to develop a relatively simple dynamicstrategy for choosing a sensing matrix the utility of which can beexpressed in closed form and that outperforms the DI alternative. Acommon approach for recovery with low-density measurement matrices is touse belief propagation via message passing, the most well-knownapplication of which is low-density parity check (LDPC) optimum errorcorrection decoding.

For LDPC (and compressive sensing methods), performance guarantees comeas asymptotic bounds on the

₂-norm, but little is known for optimal design in the finite regime. Adifficulty in the design arises from the inherent multi-hypothesistesting problem associated with sensing several resources at the sametime. This is why, to develop the dynamic design according to thepresent disclosure, a GT approach allows consideration of a binaryhypothesis test for each measurement. In this way, the complexity of theanalysis is relatively low, and the expected performance for any sensingmatrix can be derived under mild assumptions. In the model of thepresent disclosure, an uninformative prior can be assigned to theφ_(i)'s to run the belief propagation message-passing algorithm on theobtained measurements. Prior to providing more details, a remarkregarding related GT approaches is in order.

Note that in the context of GT, little is known in presence ofmeasurement errors that depend on the group size, which is the scenariothe present disclosure considers. In the model of the presentdisclosure, the false alarm and misdetection probabilities depend on theoptimization of the test threshold; therefore, the noise is notindependently added, nor can an independent dilution be considered.Furthermore, the strategy derived depends on the finite horizon for K;that is, the results are not asymptotic.

For each test a binary group test is defined as follows:

$\begin{matrix}\left\{ \begin{matrix}{H_{0}\text{:}} & {{\forall{i \in {_{k}\mspace{25mu} s_{i}}}} = 0} \\\; & {\left. \Rightarrow{\theta_{0}\lbrack k\rbrack} \right. = {b_{k}^{T}n}} \\{H_{1}\text{:}} & {{\exists{i \in {_{k}\mspace{14mu} {s.t.\mspace{14mu} s_{i}}}}} = 1} \\\; & {\left. \Rightarrow{{\theta \lbrack k\rbrack} \geq {{\left. (_{i}^{\min}b_{ki} \right)\phi_{\min}} + {b_{k}^{T}n}}} \right. = {\theta_{\min}\lbrack k\rbrack}}\end{matrix} \right. & {{EQ}.\mspace{14mu} 30}\end{matrix}$

Such a test is envisioned to be useful for a downlink transmission inwhich the access point may want to allow multiple communications at thesame time and can alert the signal units over a narrowband signalingchannel to access the spectrum.

It is important to highlight that the two hypotheses pertain exclusivelyto the group of sub-bands explored in test, that is,

_(k), not the whole spectrum. Also note that this GT approach pertainsto the design of the sensing matrix and detection algorithm and not tothe underlying observation model. The different approaches against whichthe method of the present disclosure is compared subsequently usedetection strategies that are multi-hypothesis tests.

The test can be written as follows:

$\begin{matrix}{\frac{\begin{matrix}\max & {f_{\theta {\lbrack k\rbrack}}\left( {y\lbrack k\rbrack} \right)} \\{{\theta \lbrack k\rbrack} \geq {\theta_{\min}\lbrack k\rbrack}} & \;\end{matrix}}{{{f_{\theta}}_{0}\lbrack k\rbrack}\left( {y\lbrack k\rbrack} \right)}\begin{matrix}\begin{matrix}H_{1} \\ < \end{matrix} \\\begin{matrix} > \\H_{0}\end{matrix}\end{matrix}\gamma \; k} & {{EQ}.\mspace{14mu} 31}\end{matrix}$

for which the following can be derived:

$\begin{matrix}{{\alpha \left( {b_{k},\gamma_{k}} \right)} = \left( \frac{1}{\gamma_{k}\frac{\theta_{\min}}{\theta_{0}}} \right)^{\frac{\frac{\theta_{\min}}{\theta_{0}}}{\frac{\theta_{\min}}{\theta_{0}} - 1}}} & {{EQ}.\mspace{14mu} 32} \\{{\beta \left( {b_{k},\gamma_{k}} \right)} = {1 - \left( {\alpha \left( {b_{k},\gamma_{k}} \right)} \right)^{\frac{\theta_{0}}{\theta_{\min}}}}} & {{EQ}.\mspace{14mu} 33}\end{matrix}$

The decision declares that resource i is busy (

_(l) is true) if the majority of the tests where resource i is involvedis positive, else it accepts the null hypothesis

₀ for resource i. Thus:

$\begin{matrix}{{\pi_{0}\left( {i,b,\gamma} \right)} = {{\left( {1 - {\prod\limits_{j \in {A_{k}\backslash i}}\omega_{j}}} \right)\left( {1 - {\beta_{i}\left( {b,{\gamma;0}} \right)}} \right)} + {{\alpha \left( {b,\gamma} \right)}{\prod\limits_{j \in {A_{k}\backslash i}}\omega_{i}}}}} & {{EQ}.\mspace{14mu} 34} \\{\mspace{79mu} {{\pi_{i}\left( {i,b,\gamma} \right)} = {1 = {\beta_{i}\left( {b,{\gamma;1}} \right)}}}} & {{EQ}.\mspace{14mu} 35}\end{matrix}$

where the functions π_(i)(i, b, γ), j=0, 1 are only defined whenb_(i)≠0. These functions represent the probabilities of declaring

₁ in a group-test defined by b with threshold γ and given s_(i)=j,j=0, 1. Notice that the error probabilities α, β refer to each binaryhypothesis testing defined in Equation 30. The notation for β_(i)(b, γ;s_(i)) indicates the probability of having a missed-detectionconditioned on the state s_(i) of one of the resources. It then followsthat

$\begin{matrix}{{\alpha_{i}^{GT}\left( {B,\gamma} \right)}\overset{\Delta}{=}{1 - {F_{PBD}\left( {{{\left\lceil \frac{B_{i}}{2} \right\rceil - 1};{B_{i}}},\left\{ {{\pi_{0}\left( {i,b_{k},\gamma_{k}} \right)}:{k \in B_{i}}} \right\}} \right)}}} & {{EQ}.\mspace{14mu} 36} \\{{\beta_{i}^{GT}\left( {B,\gamma} \right)}\overset{\Delta}{=}{F_{PBD}\left( {{{\left\lceil \frac{B_{i}}{2} \right\rceil - 1};{B_{i}}},\left\{ {{\pi_{1}\left( {i,b_{k},\gamma_{k}} \right)}:{k \in B_{i}}} \right\}} \right)}} & {{EQ}.\mspace{14mu} 37}\end{matrix}$

where F_(PBD)(k; n; p) indicates the cumulative distribution function ofa Poisson binomial distribution parameterized by p∈[0; 1]^(n). One canthen replace Equations 36 and 37 in Equation 15 to solve theoptimization in Equation 16, where the equivalence between the decisionrules δ and the selection of the thresholds γ is essentially the same asfor the DI case.

Notice that in order for Equations 36 and 37 to hold, each of the testsmust be independent, conditioned on the state of the resource i. This istrue if the sensing matrix, in the language used for LDPC codes, doesnot have length-4 cycles; that is, two different measurements do not mixmore than one sub-band in common. Such a condition is typically requiredfor belief propagation algorithms, for example, message passing, whichsuffer from loopy networks with short cycles.

The optimization remains extremely complex due to the complexity of thedecision space for B and the sum of an exponentially growing number ofterms for the probabilities defined in Equations 36 and 37.Nevertheless, it gives a method to evaluate the objective of theoptimization for any sensing matrix B, where the optimization over γ canbe numerically solved. Equations 36 and 37 are monotonic functions ofthe probabilities π₀, π₁ defined in Equations 34 and 35, which aremonotonic in the γ_(k) 's, and therefore a unique solution for γ exists.Next, additional constraints to Equation 16 are introduced, inparticular on the structure of B, in order to evaluate whether GTstrategy is superior to the DI approach.

Note that a maximum likelihood (ML) or a maximum a posterioriprobability estimator, for a rank-deficient sensing matrix, does notprovide optimality guarantees in terms of minimum error probability orminimum Bayesian risk. Nevertheless, for the same sensing matrix, themaximum a posteriori probability estimator is expected to outperform thebinary GT hypothesis in Equation 30 by simply adding more degrees offreedom to the decision in the κ-th dimensional space of theobservations. Therefore, the evaluation of the objective in Equation 16via Equations 36 and 37 provides a benchmark for the utility obtainablewith a more refined detection method.

1. The pairwise tests case: To start, matrices B are considered thathave the following property: each resource is sensed only one time,either directly inspected or mixed with another resource, and no testmixes more than two resources, that is, |

_(k)|≤L=2, |

|≤1 ∀k=1, . . . , κ, i=1, . . . , N. Consider the test that mixesentries i and j. According to the strategy derived at the beginning ofthe section, one can use Equations 34, 35, 36, and 37 to write out theper-time instant utility obtainable after the decision. First, fromEquation 30, without prior knowledge over φ_(i), φ_(j) other than thethreshold φ_(min), the best choice to minimize α is to set b_(i)=b_(j).This false alarm probability is referred to as α_(ij)). Therefore,similar to the DI case, one can consider binary coefficients for b_(k),that is, b_(ki)≠0→b_(ki)=1. This holds true also for the extension ofL>2 and gives implementation advantages as discussed in Section I, B.

A missed detection event in Equation 30 can occur for three differentstates of the resources i, j; an upper-bound for the corresponding missdetection probability is established by always considering θ=θ_(min) andis referred to as β_(ij,max). What is obtained is the following:

u _(ij) ^(GT)

ω_(i)ω_(j)(r _(i) +r _(j))(1−α_(ij))+[(ω_(i)(1−ω_(j))(r_(i)+ρ₁)+ω_(j)(1−ω_(i))(r_(j)+ρ_(i))+(1−ω_(i))(1−ω_(j))(ρ_(i)+ρ_(j)))]β_(ij,max)  EQ. 38

where the threshold for this test γ_(ij) has been set to maximizeEquation 38, that is,

$\begin{matrix}{\gamma_{ij} = {\frac{\omega_{i}{\omega_{j}\left( {r_{i} + r_{j}} \right)}}{{\left( {1 - \omega_{i}} \right)\left( {{\rho_{i}} - {\omega_{j}r_{j}}} \right)} + {\left( {1 - \omega_{j}} \right)\left( {{\rho_{j}} - {\omega_{i}r_{i}}} \right)}}.}} & {{EQ}.\mspace{14mu} 39}\end{matrix}$

Consider then a graph in which each resource is a vertex and the edgeweight u_(ij) between two vertices ij is the utility (per time instant)u_(ij) ^(GT) just defined. The weight of the loops u_(ii) ^(GT) aregiven by u_(i) ^(DI) in Equation 26. The problem can then be translatedinto a particular instance of a max-cut problem: picking a subset of theedges and forming a subgraph, in which each edge represents a test, tomaximize the objective in Equation 16. Formally, the equation can bewritten as follows:

$\begin{matrix}\begin{matrix}\begin{matrix}{maximize} \\ɛ\end{matrix} & {U^{GT}(ɛ)} \\{{subject}\mspace{14mu} {to}} & {{\deg \; {ɛ(i)}} \leq {i\mspace{14mu} {\forall{i \in }}}}\end{matrix} & {{EQ}.\mspace{14mu} 40} \\{where} & \; \\{{U^{GT}(ɛ)}\underset{\Delta}{=}{\left( {K - {ɛ}} \right)\left( {\sum\limits_{{ij} \in ɛ}u_{ij}^{GT}} \right)}} & {{EQ}.\mspace{14mu} 41}\end{matrix}$

and deg_(ε)(i) is the nodal degree of node i induced by the undirectedgraph

=(

, ε). It is possible to map the constraint on the nodal degree in theobjective of Equation 40 by adding a penalty for the violation of suchconstraint. This guarantees that the optimal solution will be equivalentto Equation 40, that is, no set of edges that violates the constraintcan improve the objective, and any feasible set of edges would have thesame objective in the two problems. The optimization can be rewritten asfollows:

$\begin{matrix}{{\begin{matrix}{maximize} \\ɛ\end{matrix}{U^{GT}(ɛ)}} - {M{\sum\limits_{i \in N}^{\;}\; {\mathrm{\Upsilon}\left( {\deg \; {ɛ(i)}} \right)}}}} & {{EQ}.\mspace{14mu} 42} \\{where} & \; \\{{\mathrm{\Upsilon}(n)}\overset{\Delta}{=}\left\{ \begin{matrix}0 & {{{for}\mspace{14mu} n} \leq 1} \\{n - 1} & {{{for}\mspace{14mu} n} \geq 2}\end{matrix} \right.} & {{EQ}.\mspace{14mu} 43}\end{matrix}$

and M is a positive constant.

Lemma 2:

For M>0, the objective in Equation 42 is a non-monotone sub-modularfunction of ε, and it is possible to find M*>0 such that for any M>M*the two optimizations of Equations 40 to 42 are equivalent.

The extension of this result for L>2 can now be discussed to develop ageneral algorithm that leverages the sub-modularity of the optimumdesign problem in Equation 40.

2. Extension to L>2: If more than two channels are mixed, instead ofjust edges or self-loops to indicate the tests, cycles of length can beobtained up to L. The nodal degree in Equation 42 then is interpreted asthe number of cycles to which a node belongs, and the set of edges isreplaced with the set of cycles. The set ε of edges is then replacedwith the set C of possible cycles, and use c to indicate the genericcycle, which could be a self-loop, an edge, or a cycle with length 3 orgreater. With these substitutions the proof of sub-modularity in Lemma 2naturally extends to this case as well. In light of the constraint |

_(i)|≤1, no node can be in two cycles.

Algorithm 1: Greedy Maximization of U^(GT) ( 

 ) 1: Initialize: 

 = Ø. 2: While ∃ C ∈ 

 such that ∂_(c) U^(GT)( 

 ) > 0 3:  Find c* = arg 

 ( 

 ) 4:   

 ← 

 ∪ C* 5: End

3. The factor approximation of the greedy algorithm: Having proved thesub-modularity of Equation 42 in Lemma 2, it is natural to resort to agreedy procedure; however, it is important to highlight that theobjective in Equation 42 does not respect the non-negativity property.Due to the particular structure of the problem, it is possible to find afactor approximation for the output of the greedy algorithm.

Lemma 3:

Algorithm 1 guarantees an α-constant factor approximation of the optimalsolution for Equation 42, where

$\begin{matrix}{\alpha = {\frac{1}{\min \left\{ {L_{eff},\frac{K}{2}} \right\}}{\frac{K - 1}{K - {\min \left\{ {L_{eff},\frac{K}{2}} \right\}}}.}}} & {{EQ}.\mspace{14mu} 44}\end{matrix}$

Note that

$\begin{matrix}{{\partial_{C^{\prime}}{U^{GT}{()}}} = {{\sum\limits_{C \in}^{\;}\; u_{C}} + {\left( {K - {} - 1} \right)u_{C^{\prime}}}}} & {{EQ}.\mspace{14mu} 45}\end{matrix}$

so, as long as the number of tests |C| added in the greedy maximizationis less than the time horizon K, then

arg   max C ∈ C  ∂ C  U GT  ( ) = arg   max C ∈ C  u C . EQ . 46

This relation indicates that, in the greedy procedure, edges are addedin decreasing order of utility, respecting the constraint on the nodaldegree in light of Lemma 2. Also, from Equation 45, it is easy to findthat the optimal |C| never exceeds

$\left\lceil \frac{K - 1}{2} \right\rceil.$

In the greedy procedure in Algorithm 1, there is a constant number ofoperations per query, which indicates the overall complexity of thealgorithm is dominated by the sorting of all possible cycles' utilities.In the worst case, sorting n values require O(n²) operations, thus thecomplexity is given by

${0\left( \left( {\sum\limits_{ = 1}^{L}\left( \frac{n}{} \right)} \right)^{2} \right)},$

that is, polynomial in N and exponential in L.

C. Additional Applications of the Stochastic Optimization

The analysis for the factor approximation of the greedy strategytranscends the spectrum-sensing application discussed in the presentdisclosure. Group testing has been applied to a number of disparatecontexts to model the outcome of sequential tests. As long as one has away to define the per-time utility derived from each test as in Equation38 and an overall utility as in Equation 41, then the results of thepresent disclosure can be applied. Classes of problems that can beformulated in a similar way include job scheduling for data centers,design of parity checks for rateless coding, and dynamic advertisementthat promotes an offer that bundles two products/services together, forexample. Obviously, in all these cases the statistics of theobservations are radically different.

IV. Approximate Maximum Likelihood Estimate for Mixed Tests

The previous sections have provided methods that find a low-densitymeasurement matrix. As will be apparent in the numerical results inSection V, the noise-folding phenomenon justifies the use ofsparse-sensing matrices. They are also ideal when one wants to applybelief propagation to the decision problem. However, for the sake ofcomparison, an approach is proposed here which can be applied to anymeasurement matrix B and that can be mapped into previous solutions.

Assume that K measurements have been collected, by mixing a set

⊆

of sub-bands. One could ignore the prior ω_(i) and derive the maximumlikelihood (ML) estimate for φ. The log-likelihood function is asfollows:

$\begin{matrix}{{\log \left( {f\left( y \middle| \phi_{A} \right)} \right)} = {{{- {\sum\limits_{k = 1}^{\kappa}{\log \; {\theta \lbrack k\rbrack}}}} + \frac{y\lbrack k\rbrack}{\theta \lbrack k\rbrack}}\overset{{\theta {\lbrack k\rbrack}}\rightarrow{y{\lbrack k\rbrack}}}{\approx}{{- {\sum\limits_{k = 1}^{\kappa}1}} + {\log \mspace{14mu} {y\lbrack k\rbrack}} + {\frac{1}{2}\left( \frac{{y\lbrack k\rbrack} - {\theta \lbrack k\rbrack}}{y\lbrack k\rbrack} \right)^{2}}}}} & {{EQ}.\mspace{14mu} 47}\end{matrix}$

where the linearization corresponds to the Taylor expansion of thelikelihood function around the observations mean (recall Equations 14and 15). A possible approach consists in solving the following LASSOproblem:

$\begin{matrix}{{\hat{\phi}}_{A} = {{\begin{matrix}{\arg {\mspace{11mu} \;}\min} \\\phi_{A}\end{matrix}{{\lambda_{A}\phi_{A}^{T}}}_{1}} + {\frac{1}{2}{\left( {y - {B\left( {\phi_{A}^{T} + n_{A}^{T}} \right)}} \right)}_{C^{- 1}}^{2}}}} & {{EQ}.\mspace{14mu} 48}\end{matrix}$

with C=diag(y) denoting the covariance of the observations and

the vector of weights for the weighted

₁-norm. The first penalty term in the objective enhances sparsity, whilethe second term comes from the ML estimate in Equation 47. Toincorporate the information of the prior beliefs ω_(i), one can setλ_(i)=

_(i) from Equation 20, ∀i∈

, to favor the estimates φ_(i)>0 for entries with lower thresholdsγ_(i). Alternatively, one can also set λ_(i)=λ∀i∈

.

V. Simulation Results

This section showcases the ability of the approach of the presentdisclosure to dynamically switch between a DI receiver (scanningreceiver) and a GT approach, based on the expected occupancy (the vectorof priors ω), the time available K, the minimum SNR threshold

${{SNR}_{\min} = \frac{\phi min}{w}},$

and the number of resources N. In the context of spectrum sensing (case0), the parameters r_(i) and ρ_(i) can be mapped into a maximization ofthe overall weighted network throughput: the reward r_(i) can beproportional to the achievable rate over the channel i in the absence ofprimary user communications, that is, r_(i) ∝log(1+SNR_(i,S)), where thesuffix S indicates the secondary communication, whereas the penaltyρ_(i) can be made proportional to the loss in rate caused to the primarycommunication due to the interference added by the secondary.

For the cognitive radio application, the concept of exploitation of theresource is tied to the definition of utility function chosen inEquation 15, which is expressed in bits /s/

. From Equations 15 and 16, r_(i)'s and ρ_(i)'s can be normalized overthe communication bandwidth without altering the optimization. Thelonger the time available to transmit, the larger the number of bitsthat can be transmitted over that band.

For the other case, that is, when the reward comes from detectingcorrectly which resources are busy, for example, a RADAR application, itis not immediately clear why the utility would be proportional to thenumber of remaining time instants. To interpret this, the action upondeclaration of s_(i)=1 is modelled as a Bernoulli trial that accrues areward r_(i) if such action is successful, that is, the target isactually hit, and this happens with a certain probability ρ_(i) for eachattempt. The number of attempts T_(i) necessary to hit the target isthen geometrically distributed. One can find then that the expectedreward is equal to

r _(i) P(T _(i)≤(K−κ))=r _(i)Σ_(k−1) ^(K-κ)ρ_(i)(1−p _(i))^(k−1) =r_(i)(1−(1−p _(i))^(K-κ))≈(K−κ)r _(i) p _(i)

for small p_(i), which would motivate having an expected utility thatincreases linearly with time. The ρ_(i) associated with this case modelsan intervention cost, the main purpose of which is to limit the falsealarm rate.

It is important to highlight, however, that the time dependency in theoptimization objective prevents the formulation from returning to astandard constant false alarm rate (CFAR) detection method.Nevertheless, the model can apply to electronic warfare (tentatives ofcreating jamming), wake-up radio, and other problems where the action(and the associated utility) is on the channels that are declared busy.For all the figures, reference is made to L=2, 3 as the maximum numberof resources per test allowed in the greedy procedure in Algorithm 1.Theoretically, the optimal value for U^(GT) monotonically increases withL since increasing L introduces additional degrees of freedom. However,the simulations used the greedy solution and, as proved in Lemma 3, theapproximation factor of the greedy maximization is potentially worse forhigher values of L, as the following numerical results show.

“Group Testing” indicates the utility obtained with the GT approach. Themaximum a posteriori probability estimator (MAP estimator) is theestimator that knows the true values Φ_(i), uses the same matrix B ofthe GT approach, but then decides on each resource based on theposterior for ω_(i), using belief propagation.

1. Spectrum sensing vs. RADAR: Even though in light of the symmetry inthe definition of the threshold γ_(i) one can switch the r's and ρ's togo from spectrum sensing (case 0) to RADAR (case 1) and find the sametrends, even for the combined tests, to avoid confusion, the differencein the two scenarios is highlighted in the first simulation presented inFIGS. 4A and 4B.

FIG. 4A is a graphical representation comparing utility for differentoptimization approaches for the spectrum sensing application. FIG. 4B isa graphical representation comparing utility for different optimizationapproaches for the RADAR application.

For the experiment in FIGS. 4A and 4B, the following were set−K=30,N=60, and r_(i)=r, ρ_(i)=ρ, and ω_(i)=ω, SNR_(i)=SNR_(min)(10 dB) ∀i−

—so that we have that for ω equal to

$\frac{\rho}{\rho + r}\mspace{14mu} {or}\mspace{14mu} \frac{r}{\rho + r}$

for case 0 and case 1, respectively. These are the threshold valuesgiven in Assumption 1 to guarantee that no resource can give positiveutility if not tested. In both scenarios the utility increases with theratio

$\frac{\rho}{r},$

since the prior probability that favors a positive utility functionincreases as well for both the spectrum-sensing and RADAR applications.

However, the gain for the GT approach over the DI approach happens incomplementary ranges: when

$\frac{\rho}{r} > 1$

for the spectrum-sensing application and when

$\frac{\rho}{r} < 1$

in the RADAR application. When the penalty increases with respect to thereward, the GT approach for spectrum sensing is conservative and doesnot transmit in any of the channels in a group that tested positively;nevertheless, as the priors ω_(i) increase, it is possible to findmultiple empty sub-bands with just one test and gain in utility comparedwith the DI approach.

For the RADAR application, when the penalty increases with respect tothe reward, there is a disadvantage in declaring as busy all theelements in the test, even if the prior ω decreases. Clearly, thislimits the benefit of combined tests, whereas when

$\frac{\rho}{r}$

decreases, there is a gain since one element found busy in the poolguarantees higher reward. Apart from this asymmetry, both cases show thesame trends in utility over the number of available resources N, and thevalue of SNR_(min).

2. Utility for different N: FIG. 5A is a graphical representationcomparing utility for different optimization approaches versus a ratiohorizon K over a number of resources N with the horizon K equal to 10.FIG. 5B is a graphical representation comparing utility for differentoptimization approaches versus the ratio horizon K over the number ofresources N with the horizon K equal to 30.

FIGS. 5A and 5B thus plot the utility (normalized over K²) over theratio

$\frac{K}{N}$

for two different horizons, that is, K=10 and K=30 and SNR_(min)=10 dB.Only for

${\frac{K}{N}\overset{<}{\approx}0.75},$

the GT approach outperforms all competing options, whereas when thehorizon K increases, almost no benefit comes from mixing resources. Thissuggests that there is enough time to test them independently with highaccuracy. This experiment looked at case 0 and set

${\omega_{i} \sim \left( {0.7,\frac{\rho \; i}{\rho_{i} + r_{i}}} \right)},{where}$r_(i) = log (1 + SNR_(i, S))  and  ρ_(i) = 5r_(i)  with  SNR_(i, S_(dB)) ∼ ([10, 20]).

The SNR for the test,

$\frac{\phi_{i}}{n_{i}},$

is generated uniformly between 10 dB and 20 dB; recall that the onlyinformation used in the algorithm is the minimum SNR value, that is, inthis case 10 dB. In the regime considered, the DI approach isapproximately constant since it is easy to show

$U^{{DI},{OPT}} \leq {\frac{K^{2}}{4}u_{\max}}$

irrespective of N.

3. Utility for different SNR_(min): FIG. 6A is a graphicalrepresentation comparing utility for different optimization approachesversus the minimum SNR with the horizon K equal to 30. FIG. 6B is agraphical representation comparing utility for different optimizationapproaches versus the minimum SNR with the horizon K equal to 10. Thisset of experiments studied how the utility behaves versus the minimumSNR in each active sub-band. In this case the SNR was drawn uniformlybetween SNR_(min) _(dB) and SNR_(min) _(dB) +10, and once again only thevalue of

${SNR}_{\min} = \frac{\phi_{\min}}{w}$

was used in the optimization, which is shown in the abscissa of thefigures. Matching intuition, the GT approach outperformed the DIapproach only when SNR_(min) was sufficiently high and the gain inutility was larger for K=10 than for K=30.

For this experiment the number of resources was fixed to N=20 and, aspreviously highlighted, increasing K for fixed N diminishes the benefitof combining resources in a test. In this case the utility was alsoplotted that was obtainable with the approximate ML estimate obtainedvia compressive sensing, described in Section IV. For this case, toillustrate the noise-folding issue, a dense matrix that had the sameaspect ratio of the one found via the GT approach was used, that is, onethat scans the same set of resources for the same number of tests.

To show reasonable results, only for the ML estimate via compressivesensing, the mean of y[k] over 10 samples was taken. Despite having moremeasurements, such approach gives a much lower utility than the DI andthe GT approach of the present disclosure due to the negative effect ofnoise folding. For K=30, the approach of the present disclosure was alsocompared with the performance obtained using belief propagation in aloopy network and an LDPC matrix. Considering N=20 resources and anexpected sparsity approximately equal to 4, a regular LDPC matrix waschosen with a row weight of 5 and a column weight of 3, resulting in 12tests. The LDPC was not implemented for K=10 since the constraints onthe regularity would have given either a diagonal matrix (same as DI) ora relatively dense matrix. The absence of any optimization in the choiceof which and how many resources to test produces a utility that, for lowSNR, is lower than the DI approach proposed. For a high enough SNR, theLDPC design can outperform the DI approach but still gives a utilitylower than the GT strategy with L=2. This highlights the benefit ofhaving an active sub-Nyquist receiver compared with a static offlineselection of the parameters.

The various illustrative logical blocks, modules, and circuits describedin connection with the embodiments disclosed herein may be implementedor performed with a processor, a digital signal processor (DSP), anapplication specific integrated circuit (ASIC), a field programmablegate array (FPGA), or other programmable logic device, a discrete gateor transistor logic, discrete hardware components, or any combinationthereof designed to perform the functions described herein. Furthermore,a controller may be a processor. A processor may be a microprocessor,but in the alternative, the processor may be any conventional processor,controller, microcontroller, or state machine. A processor may also beimplemented as a combination of computing devices (e.g., a combinationof a DSP and a microprocessor, a plurality of microprocessors, one ormore microprocessors in conjunction with a DSP core, or any other suchconfiguration).

The embodiments disclosed herein may be embodied in hardware and ininstructions that are stored in hardware, and may reside, for example,in RAM, flash memory, ROM, electrically programmable ROM (EPROM),electrically erasable programmable ROM (EEPROM), registers, a hard disk,a removable disk, a CD-ROM, or any other form of computer-readablemedium known in the art. An exemplary storage medium is coupled to theprocessor such that the processor can read information from, and writeinformation to, the storage medium. In the alternative, the storagemedium may be integral to the processor. The processor and the storagemedium may reside in an ASIC. The ASIC may reside in a remote station.In the alternative, the processor and the storage medium may reside asdiscrete components in a remote station, base station, or server.

It is also noted that the operational steps described in any of theexemplary embodiments herein are described to provide examples anddiscussion. The operations described may be performed in numerousdifferent sequences other than the illustrated sequences. Furthermore,operations described in a single operational step may actually beperformed in a number of different steps. Additionally, one or moreoperational steps discussed in the exemplary embodiments may becombined. Those of skill in the art will also understand thatinformation and signals may be represented using any of a variety oftechnologies and techniques. For example, data, instructions, commands,information, signals, bits, symbols, and chips, that may be referencesthroughout the above description, may be represented by voltages,currents, electromagnetic waves, magnetic fields, or particles, opticalfields or particles, or any combination thereof.

Unless otherwise expressly stated, it is in no way intended that anymethod set forth herein be construed as requiring that its steps beperformed in a specific order. Accordingly, where a method claim doesnot actually recite an order to be followed by its steps, or it is nototherwise specifically stated in the claims or descriptions that thesteps are to be limited to a specific order, it is in no way intendedthat any particular order be inferred.

Those skilled in the art will recognize improvements and modificationsto the preferred embodiments of the present disclosure. All suchimprovements and modifications are considered within the scope of theconcepts disclosed herein and the claims that follow.

What is claimed is:
 1. A radio frequency (RF) receiver, comprising: adynamic modulator configured to modulate a received signal such that oneor more portions of the received signal are dynamically selected andfolded into a baseband of a modulated signal; an energy detectorconfigured to sample an energy of the modulated signal at a sub-Nyquistrate; and a controller configured to adjust the dynamic modulator basedon an output of the energy detector.
 2. The RF receiver of claim 1,wherein the output of the energy detector is a past sample of the energyof the modulated signal.
 3. The RF receiver of claim 1, wherein thedynamic modulator is further configured to modulate the received signalwith a mixture of sinusoidal components whose frequencies and amplitudesare dynamically adjusted by the controller across multiple time windowsto select the one or more portions of the received signal.
 4. The RFreceiver of claim 1, wherein the energy detector samples the energy ofthe modulated signal at the sub-Nyquist rate by low-pass filtering andsampling the baseband of the modulated signal.
 5. The RF receiver ofclaim 1, wherein the dynamic modulator comprises a set ofvoltage-controlled oscillators (VCOs).
 6. The RF receiver of claim 5,wherein a voltage level of each of the set of VCOs is proportional to acoefficient of a sensing matrix output by the controller, therebygenerating a sinusoidal signal around a selected frequency.
 7. The RFreceiver of claim 1, wherein the dynamic modulator comprises a pluralityof fixed frequency oscillators, each fixed frequency oscillator beingcontrolled by the controller.
 8. The RF receiver of claim 1, wherein theenergy detector comprises: a low pass filter coupled to an output of thedynamic modulator; and a sampling circuit coupled to an output of thelow pass filter.
 9. The RF receiver of claim 8, wherein the samplingcircuit is configured to sample the modulated signal after the low-passfilter at intervals based on a bandwidth of the baseband.
 10. A methodfor sensing occupied radio frequency (RF) spectrum, comprising:receiving an RF signal; sequentially modulating the RF signal using asensing matrix to produce a set of modulated signals; and detecting asignal occupancy for each of the set of modulated signals; whereincoefficients of the sensing matrix are dynamically adjusted based ondetecting the signal occupancy.
 11. The method of claim 10, wherein:each of the set of modulated signals corresponds to an RF resource; andthe coefficients of the sensing matrix are dynamically adjusted based oncorrectly sensing occupancy of each RF resource.
 12. The method of claim11, wherein: each RF resource comprises an RF channel; and the set ofmodulated signals is sequentially combined in a baseband signal prior todetecting the signal occupancy.
 13. The method of claim 10, furthercomprising: predicting a plurality of unoccupied RF channels based onthe signal occupancy; and transmitting an output RF signal over theplurality of unoccupied RF channels.
 14. The method of claim 10, furthercomprising: determining an occupied RF channel based on the signaloccupancy; and tuning to the occupied RF channel to convert content ofthe occupied RF channel from analog to digital.
 15. An analog front endfor a radio frequency (RF) receiver, comprising: a dynamic modulatorconfigured to convert an incoming RF signal having a first bandwidth toa modulated signal having content of the first bandwidth dynamicallyfolded in a second bandwidth narrower than the first bandwidth; and anenergy detector configured to sense signal energy of the modulatedsignal within the second bandwidth.
 16. The analog front end of claim15, wherein: the second bandwidth defines a baseband; and the modulatedsignal comprises spectral content of the incoming RF signal folded intothe baseband.
 17. The analog front end of claim 16, wherein the dynamicmodulator is configured to convert the incoming RF signal by: selectinga set of sub-bands of the first bandwidth for the energy detector tosense; and folding the set of sub-bands into the second bandwidth. 18.The analog front end of claim 17, wherein the set of sub-bands isselected as a function of a past output of the energy detector.
 19. Theanalog front end of claim 16, further comprising a controller configuredto determine occupancy of a set of RF channels based on the energydetector sensing signal energy above a threshold.
 20. The analog frontend of claim 19, wherein the controller is further configured to adjustthe dynamic modulator based on the occupancy of the set of RF channels.